58 Chapter 3: WL/WAL Crossover and <strong>Sp<strong>in</strong></strong> Relaxation <strong>in</strong> Conf<strong>in</strong>ed Systems all wave vectors, K = Q/Q SO , is given by with E B,2D,1 /D e Q 2 SO = 1+K 2 x, (3.100) E B,2D,2 /D e Q 2 SO = E B,2D,1 + f 1 E B,2D,3/4 /D e Q 2 SO = E B,2D,1 − 1 2 + 1 6 f 1/3 2 − 1 3 f1/3 2 , (3.101) ( 1±i √ 3 ) f1 f 1/3 2 ( 1∓i √ 3 ) f 1/3 2 , (3.102) f 1 = ˜B 2 −4Kx 2 −1, (3.103) ( √3 √ ) f 2 = 3 108Kx 4 +f3 1 −18K2 x , (3.104) ˜B = gµ B B/D e Q 2 SO, (3.105) Thus, there are sp<strong>in</strong> states with the same real part <strong>of</strong> the Cooperon energy, so that they decay equally <strong>in</strong> time, but the imag<strong>in</strong>ary part is different, so that they precess with different frequencies around the magnetic field axis. A significant change <strong>of</strong> the Cooperon spectrum appears when gµ B B/D e exceeds Q 2 SO , as can be seen <strong>in</strong> Fig.3.16(b). All states with a low decay rate do precess now, due to a f<strong>in</strong>ite imag<strong>in</strong>ary value <strong>of</strong> their eigenvalue. Associated with this change is also a change <strong>of</strong> the dispersion <strong>of</strong> the real part <strong>of</strong> E B,2D,3 which changes for K y = 0 from a nearly quadratic dispersion <strong>in</strong> K x , a 0 + a 1 K 2 x for ˜B < 1 to one which changes more slowly as a 0 +a 1 K 2/3 x +a 2 K 4/3 x +a 3 K 2 x for ˜B ≥ 1 [see Fig.3.16 (a)]. Weak Field In the case <strong>of</strong> a weak Zeeman field, ˜B ≪ 1, the s<strong>in</strong>glet and triplet sectors are still approximately separated. A f<strong>in</strong>ite ˜B ≪ 1 lifts however the energy <strong>of</strong> the s<strong>in</strong>glet mode to E B,2D,2 (K = 0)/D e Q 2 SO = ˜B 2 /2 + O(˜B 4 ), thus the s<strong>in</strong>glet mode atta<strong>in</strong>s a f<strong>in</strong>ite gap, correspond<strong>in</strong>g to a f<strong>in</strong>ite relaxation rate. The absolute m<strong>in</strong>imum <strong>of</strong> two <strong>of</strong> the triplet modes is also lifted by E B,2D,2 (K = ± √ 15/4)/D e Q 2 = 7/16 + (3/4) ˜B 2 SO + O(˜B 4 ), while their value is <strong>in</strong>dependent <strong>of</strong> ˜B at K = 0. In contrast, the m<strong>in</strong>imum <strong>of</strong> the triplet mode E B,2D,3 , whichapproachesE T+ <strong>in</strong>thelimit<strong>of</strong>nomagneticfield(seeFig.3.4)isdim<strong>in</strong>ishedto E B,2D,3 (K = 0)/D e Q 2 = 2− ˜B 2 SO /2+O(˜B 4 ). So, <strong>in</strong> summary, a weak Zeeman field renders all four Cooperon modes gapfull and that gap can be <strong>in</strong>terpreted as a f<strong>in</strong>ite relaxation rate
Chapter 3: WL/WAL Crossover and <strong>Sp<strong>in</strong></strong> Relaxation <strong>in</strong> Conf<strong>in</strong>ed Systems 59 3.0 2.5 R[E]/DeQ 2 SO 2.0 1.5 1.0 0.5 0.0 2 1 0 1 2 0.3 0.2 (a) K x I[E]/DeQ 2 SO 0.1 0.0 0.1 0.2 0.3 2 1 0 1 2 (b) K x Figure 3.15: (a) Real and (b) imag<strong>in</strong>ary parts <strong>of</strong> the spectrum <strong>of</strong> the 2D Cooperon with Zeeman term <strong>of</strong> strength gµ B B/D e Q 2 SO = 0.4. E B,2D,1 (black), E B,2D,2 (red dashed), E B,2D,3 (green), E B,2D,4 (blue dashed). Dashed vertical l<strong>in</strong>es are located at K x = ±1/ √ 2, the wave vector where the triplet mode E T− and the s<strong>in</strong>glet mode E S are cross<strong>in</strong>g each other (without loss <strong>of</strong> generality K y = 0).